Make the simple syrup: Whisk erythritol and xanthan gum together until combined. Add water to a saucepan, whisk in erythritol mixture. Bring to a boil whisking continuously. Once boiling, lower heat slightly but maintain a boil. Do not whisk once boiling - use a wet pastry brush to wipe down any crystals forming on the inside of the pan. Boil until the mixture is completely clear, a few minutes. Remove from heat, stir in lemon zest and basil. Cover and steep until cool. Strain into a tight-lidded jar. Keeps refrigerated for at least a week.

To make the cocktail: Combine gin, simple syrup, and lemon juice in a shaker with ice. Shake briskly.

Strain into a martini glass. Gently pour the Tennessee Honey over the back of a spoon to float it on top. Garnish and enjoy.